Search of new physics through flavor physics observables

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Olcyr Sumensari

To cite this version:

Olcyr Sumensari. Search of new physics through flavor physics observables. High Energy Physics -Phenomenology [hep-ph]. Université Paris Saclay (COmUE), 2017. English. �NNT : 2017SACLS315�. �tel-01739406�

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Search of new physics through

flavor physics observables

Thèse de doctorat de l'Université Paris-Saclay préparée à l’Université Paris-Sud

École doctorale n°564 Physique en Ile de France (EDPIF) Spécialité de doctorat : Physique

Thèse présentée et soutenue à Orsay, le 27 septembre 2017, par

M. Olcyr Sumensari

Composition du Jury :

Mme Asmaa Abada

Professeur, Université Paris-Sud (UMR 8627) Présidente

M. Antonio Pich

Professeur, Université de Valencia Rapporteur

M. Gino Isidori

Professeur, Université de Zurich Rapporteur

M. Diego Guadagnoli

Chargé de recherche, LAPTh (UMR 5108) Examinateur

Mme Justine Serrano

Chargée de recherche, CPPM (UMR 7346) Examinatrice

Mme Renata Zukanovich Funchal

Professeur, Université de São Paulo Examinatrice

M. Damir Bečirević

Directeur de recherche, Université Paris-Sud (UMR 8627) Directeur de thèse

NNT : 2 0 1 7 S A CL S 3 1 5

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Université Paris-Saclay

Espace Technologique / Immeuble Discovery

Titre : Recherche de la nouvelle physique à travers les observables de la physique de la

saveur

Mots clés : changement de la saveur, B meson, désintégration semileptonique, théorie

effective des champs, Higgs, leptoquark.

Résumé : La recherche indirecte des effets

de la physique au-delà du Modèle Standard à travers les processus de la saveur est complémentaire aux efforts au LHC pour observer directement la nouvelle physique. Dans cette thèse nous discutons plusieurs scénarios au-delà du Modèle Standard (a) en utilisant une approche basée sur les théories de champs effective et (b) en considérant des extensions explicites du Modèle Standard, à savoir les modèles à deux doublets de Higgs et les scénarios postulant l'existence des bosons leptoquarks scalaires à basse énergie.

En particulier, nous discutons le phénomène de la brisure de l'universalité des couplages leptoniques dans les désintégrations basées sur les transitions 𝑏 → 𝑠𝜇𝜇 and 𝑏 → 𝑐𝜏𝜈, et la possibilité de chercher les signatures de la violation de la saveur leptonique à travers les modes de désintégration similaires. Une proposition pour tester la présence d'un boson pseudoscalaire léger à travers les désintégrations des quarkonia est aussi présentée.

Title : Search of new physics through flavor physics observables

Keywords : flavor changing, B meson, semileptonic decay, effective field theory, Higgs

particle, leptoquark.

Abstract : Indirect searches of physics

beyond the Standard Model through flavor physics processes at low energies are complementary to the ongoing efforts at the LHC to observe the New Physic phenomena directly. In this thesis we discuss several scenarios of physics beyond the Standard Model by (a) reusing the effective field theory approach and (b) by considering explicit extensions of the Standard Model, namely the two-Higgs doublet models and the scenarios involving the low energy scalar leptoquark states.

Particular emphasis is devoted to the issue of the lepton flavor universality violation in the exclusive decays based on 𝑏 → 𝑠𝜇𝜇 and 𝑏 → 𝑐𝜏𝜈, and to the possibility of searching for signs of lepton flavor violation through similar decay modes. A proposal for testing the presence of the light CP-odd Higgs through quarkonia decays is also made.

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Remerciements 4

Introduction 6

1 The Standard Model 9

The Standard Model 9

1.1 The Standard Model Lagrangian . . . 9

1.2 Why do we need to go beyond? . . . 14

1.2.1 Neutrino masses. . . 14

1.2.2 The SM flavor problem . . . 16

1.2.3 The hierarchy problem . . . 18

2 Flavor observables as a probe of New Physics 20 Flavor observables as a probe of New Physics 20 2.1 Effective field theories . . . 21

2.2 Tree-level electroweak decays of mesons . . . 21

2.2.1 Leptonic decays of mesons . . . 23

2.2.2 Semileptonic P → P0 _decays _{. . . 24}

2.2.3 Brief discussion of P → V semileptonic decays . . . 30

2.2.4 Phenomenological analysis . . . 31 2.3 FCNC processes . . . 38 2.3.1 Bs→ `+`− . . . 39 2.3.2 B → K`+_`− _{. . . 40} 2.3.3 B → K∗_`+_`− _{and B} s → φ`+`− . . . 41

2.4 Lepton flavor violation in lepton decays . . . 43

2.4.1 ` → `0_γ _{and neutrino masses} _{. . . 43}

2.4.2 Status of current LFV searches . . . 45

2.4.3 Sterile neutrinos and LFV . . . 46

2.5 LFV in b → s exclusive decays . . . 51 2.5.1 Bs→ `1`2 . . . 52 2.5.2 B → K`1`2 . . . 53 2.5.3 B → K∗_` 1`2 and Bs → φ`1`2 . . . 54 2.5.4 Numerical significance . . . 59

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Scrutinizing Two-Higgs doublet models 62

3.1 General aspects of 2HDM . . . 63

3.1.1 The extended scalar sector . . . 63

3.1.2 Including fermions . . . 65

3.1.3 Model spectrum and theory constraints . . . 67

3.2 Phenomenology of heavy scalars . . . 70

3.3 Leptonic and semileptonic decays of mesons . . . 72

3.4 Wilson coefficients for the b → s`` transition . . . 74

3.5 Lessons from b → s exclusive observables . . . 83

3.6 The light CP-odd Higgs window . . . 88

3.6.1 General scan for a light CP-odd . . . 88

3.7 Seeking the CP-odd Higgs via h → P `` . . . 90

3.8 Probing a light CP-odd Higgs through quarkonia decays . . . 97

4 Leptoquarks at low and high energies 103 Leptoquarks at low and high energies 103 4.1 Introduction . . . 103

4.1.1 SU(5) unification and light leptoquarks . . . 104

4.1.2 Proton stability . . . 106

4.2 A concrete SU(5) realization with light leptoquarks . . . 108

4.2.1 Model setup . . . 109

4.2.2 RGE equations . . . 110

4.2.3 Scan of parameters . . . 111

4.2.4 Unification scale and proton decay . . . 112

4.3 Specific leptoquark states. . . 115

4.4 Direct searches at the LHC. . . 121

5 Lepton flavor (universality) violation 124 Lepton flavor (universality) violation 124 5.1 Introduction . . . 124

5.2 Lepton flavor (universality) violation in b → s`1`2 . . . 127

5.2.1 Effective description of the deviations . . . 127

5.2.2 Proposed explanations of RK and RK∗ . . . 133

5.2.3 Tree-level leptoquark models for RK(∗) . . . 135

5.2.4 Loop induced leptoquark models: A first attempt . . . 136

5.2.5 A viable leptoquark explanation of RK(∗) <1 through loops . . . 140

5.3 Lepton flavor universality violation in b → c`ν . . . 149

5.3.1 Standard Model predictions for RD and RD∗ . . . 150

5.3.2 Simultaneous explanations of RK(∗) and R_D(∗) . . . 153

5.3.3 (3, 2)1/6 and light right-handed neutrinos . . . 155

Conclusion 167 Publication List 170 A Useful identities 172 A.1 Fierz identities . . . 172

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B 2HDM 174

B.1 Feynman rules for 2HDM. . . 174

B.2 Auxiliary functions . . . 175

B.2.1 Wilson Coefficients for the Derivative Operators . . . 176

B.2.2 Wilson Coefficients Suppressed by m` . . . 177

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Je voudrais tout d’abord remercier Sébastien Descotes-Genon, le directeur du Laboratoire de Physique Théorique d’Orsay, pour son accueil au sein du laboratoire et pour tout son soutien au cours de ces trois années. Je remercie aussi les autres membres administrat-ifs et scientifiques du laboratoire, toujours très attentadministrat-ifs et disponibles pour m’aider. Je garderai de très bons souvenir de l’ambiance agréable mais sérieuse de notre laboratoire. Je pense surtout à Asmaa Abada, toujours très gentille, qui m’a permis de participer à plusieurs événements du réseau Invisibles/Elusives et avec qui j’ai pu aussi collaborer. Merci également pour la recette d’infusion à gingembre qui a sauvé un vulnérable brésilien pendant les saisons froides à Orsay.

Je tiens à remercier tout particulièrement les membres de mon jury de thèse qui ont gentiment accepté notre invitation et qui ont fait l’effort de se déplacer pour la soutenance de thèse. Je remercie tout d’abord Asmaa Abada d’avoir présidé le jury. Merci à Antonio Pich et Gino Isidori d’avoir accepté d’être rapporteurs et d’avoir lu mon manuscrit dans un court délai. Merci à Diego Guadagnoli et Justine Serrano pour leur présence et pour les discussions lors de la soutenance. C’était un grand plaisir et honneur de vous avoir tous dans mon jury de thèse.

Je suis pleinement conscient que j’étais un doctorant (doublement) privilégié. Damir Beˇcirević était un brillant directeur de thèse. Toujours très attentif, je tiens à remercier pour son immense générosité et pour avoir toujours veillé à ma croissance. C’était un énorme plaisir de partager mon temps avec une personne si gentille, si lucide, avec une si vaste connaissance en physique. Cette page est certainement trop courte pour décrire tout ce que j’ai appris de lui, sur la physique surtout, mais pas seulement. Je remercie également Renata Zukanovich Funchal, quelqu’un que je connais depuis plus de huit ans. Merci de m’avoir ouvert sa porte à l’Université de São Paulo, ma première maison. Merci pour tous les conseils, pour les nombreuses discussions et pour le constant partage de ta passion pour la science. Ton enthousiasme pour la physique est toujours pour moi une source d’inspiration. J’espère que les travaux que nous (trois) avons fait ensemble ne seront que le début d’enrichissantes collaborations.

Je tiens à remercier Pierre Fayet, mon professeur à l’Ecole polytechnique et au M2 de physique théorique, pour ses cours passionnants et pour son orientation qui m’a acheminé vers la physique des particules et, plus particulièrement, vers le LPT d’Orsay. Je remercie aussi Adel Bilal, coordinateur au M2 de physique théorique à l’ENS, pour ses excellents cours de QFT et pour ses recommandations lors de la candidature pour la thèse.

Pendant ces trois années, j’ai pu aussi travailler avec des physiciens exceptionnels. Merci à Federico Mescia pour avoir partagé ses astuces (analytiques et numériques) et pour les blagues inoubliables, bien que parfois difficiles à comprendre. Merci à Nejc Kosnik pour les nombreuses discussions à propos des leptoquarks (et les signes dans les formules !). Un grand merci à Svjetlana Fajfer pour toute sa gentillesse et pour les très plaisantes et fructueuses discussions/collaborations qui ne font que commencer. Je remercie aussi mes autres collaborateurs au Brésil, en Croatie et au Japon.

Je remercie tous les doctorants et stagiaires avec qui j’ai pu partager ces trois ann´ees : Andrei, Antoine, Florent, Florian, Gabriel, Herm`es, Luca, Lucho, Ma´ıra, Matias,

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Math-ias, Michele, Renaud, Timothé... Je garderai de très bons souvenirs de cette période (on s’est bien amusés !). Je remercie également mes amis dans le cercle polytechnicien qui ont contribué à ce que ces années en France soient si agréables : André, Anne-Sophie, Arthur, Daniel, Hudson, Juan, La´ıs, Lucas, Marlen, Nicolas, Noémie, Pedro, Rafael, Ricardo, Thi-ago, Victor... Je pars de Paris avec une “tristeza distra´ıda”. Un grand merci aussi à mes amis qui se trouvent de l’autre côté de l’Atlantique : Bruno, Eduardo, Guilherme, Leonardo, Nathália et Ricardo.

Por fim, não posso deixar de agradecer aos meus pais por todo o suporte e carinho. Mais que isso, sou eternamente grato por terem sempre me deixado tra¸car os meus próprios caminhos – sem imposi¸cões, talvez com uma dolorosa lucidez. Tudo o que eu conquistei, não somente durante o meu doutorado, come¸ca em vocês. Também agrade¸co ao meu irmão, Caio, por ter sido sempre um bom companheiro e por ter zelado por nossa mãe quando a distância era grande demais.

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The Standard Model (SM) of particle physics is a quantum gauge theory which describes with elegance and precision the interactions of sub-atomic particles. The phenomenolog-ical successes of the SM are numerous, including extensive tests at accelerator facilities performed at low and high energies which agree very well with the SM predictions. The discovery of its last missing piece at the Large Hadron Collider (LHC), the Higgs boson, represents one of its greatest achievements and corroborates six decades of persistent phe-nomenological success [1,2].

However, it is well known that the SM cannot be the ultimate theory of nature. Firstly, it does not incorporate the gravitational interaction, even though the quantum effects from gravity become significant only at inaccessibly high energies near the Planck scale. Fur-thermore, neutrinos are massless in the SM, in disagreement with the well established experimental observations that neutrinos are massive and oscillate among different flavors. There is also a growing number of astrophysical and cosmological evidences, based on the standard model of cosmology and on general relativity, which suggest that a substantial part of the matter in the universe is neither baryonic nor luminous. If one assumes the validity of those theories at large scales, then particle physics should be able to propose a new particle, which might interact by means not described by the SM of particle physics.

In addition to the experimental observations described above, which cannot be accounted for by the SM, there are also several conceptual problems which cannot be fully understood without introducing physics beyond the SM. While the gauge sector of the SM is surprisingly simple and predictive, our current understanding of flavor is highly unsatisfactory. Fermions appear in three similar copies which are only distinguished by the Yukawa interactions. Contrary to the gauge sector, the Yukawa sector is poorly constrained by symmetry. As a consequence, many parameters (fermion masses and mixing) must be fixed by confronting theory and experiment. Measurements have revealed a strong hierarchy for fermion masses and a strikingly hierarchical structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, suggesting the existence of unknown underlying symmetry still to be unveiled. The lack of understanding of the SM flavor structure is known as the flavor problem. Furthermore, quantum corrections to the Higgs boson mass are quadratically divergent, thus “unnaturally” sensitive to the ultraviolet (UV) completion of the theory. This issue is known as the hierarchy problem and it is the main motivation to searching the New Physics (NP) effects at the TeV scale.

All of the above problems call for physics beyond the SM, and many proposals have been made over the past several decades to address each of the above-mentioned issues. Simplicity and beauty have been the main guidelines in the quest for NP. However, despite the intense theoretical effort, there is no strong theoretical preference for a specific scenario of physics beyond the SM. The community of theoretical physicists has found itself at a crossroad, and it becomes necessary to use the modern day high energy experiments to select among the various options for a NP scenario. The search of NP effects can proceed via two complementary approaches: the direct searches of new particles at high energy

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facilities, and the indirect searches for NP effects in low energy observables, which will be the main focus of this thesis.

The indirect searches, and most particularly flavor physics, have been extensively used in the past to probe the high energy scales through low energy experiments. A notable example was the first observation of B0_{− B}0 mixing [3], which indicated that the top quark is much heavier than any other SM fermion years before its discovery at Tevatron [4,5]. Furthermore, the flavor physics observables provide very useful information about physics beyond the SM. A remarkable example is the K0 _{− K}0 mixing parameter K which, after comparing the SM prediction with its measured values, sets a lower bound of about 108 _{GeV for the} scale of NP under the assumptions of O(1) flavor-universal couplings [6]. In particular, this means that NP models with particles in the TeV range, as suggested by the hierarchy problem, must have a non-trivial flavor structure to comply with the stringent limits from flavor changing processes. In what concerns the indirect searches of NP, one should rely on generic approaches with the least number of assumptions as possible. Effective field theories (EFT) are the most efficient approach in that respect, since they provide a general description of low energy physics without having to postulate what happens at arbitrarily high energy scales. Another complementary approach is to consider minimal and pragmatic extensions of the SM, preferably generic, which allow us to use the information from flavor observables to guide the direct searches for new resonances at high energies. Such an example is of two-Higgs doublet model (2HDM), also embedded in various supersymmetric (or not) extensions of the SM [7]. Another possibility that became popular in recent years is to consider the various leptoquark (LQ) states, which can arise in grand unification scenarios and composite Higgs models, among other NP scenarios [8].

In this thesis we will extract the information on NP from the flavor physics observ-ables. Current experiments at NA62, BES-III and LHCb provide us with a rich set of data for testing the various NP scenarios and to guide the theoretical effort towards a flavor theory beyond the SM. The information extracted from this data will be further corrob-orated/complemented by the future experiments at Belle-II, KOTO, TREK, (g − 2)µ and Mu2E, producing a prolific scenario for flavor physics. To interpret these results, we will formulate effective theories which will be matched onto minimal models of NP motivated by the recent experimental findings. The first part of this thesis is devoted to the Higgs boson, which was the last ingredient of the SM observed in experiments. While the Higgs boson couplings measured at the LHC still allow for large deviations from the SM predic-tions, the direct searches have not ruled out the possibility of other light scalar particles in the spectrum. In this first part of the thesis, we will focus on the 2HDM, and explore the lessons on their spectrum that can be learned by using the general theoretical and phe-nomenological constraints. The scalar masses and couplings allowed by our analysis will then be confronted with the flavor physics observables. To that purpose we will compute the full set of Wilson coefficients contributing to the relevant tree-level and loop induced meson decays and confront these results with recent data. Particular emphasis will be de-voted to the exclusive b → s`+_`− _{decay modes due to the increasing experimental effort} to measure the corresponding observables at LHCb. Among the scenarios we consider, the intriguing possibility that a light CP-odd Higgs (mA. 125 GeV) is present in the spectrum will be explored. Such a particle would be most welcome as a mediator between the SM and the dark sector because the so-called pseudoscalar portal can evade the strong constraints coming from the the null results in dark matter direct detection experiments [9,10]. We will show that this scenario is perfectly plausible in light of current theoretical and experimental constraints, and we will discuss the opportunities to look for this particle in Higgs exclusive decays and in the decay modes of quarkonia.

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The second part of this thesis is devoted to the hints of lepton flavor universality (LFU) violation in semileptonic B meson decays. More specifically, the LHCb measurement of

RK = B(B → Kµ+µ−)/B(B → Ke+e−) [11] and RK∗ = B(B → K∗µ+µ−)/B(B →

K∗e+_e−_{) [}₁₂_{] in different bins of dilepton squared momentum q}2 were shown to be signifi-cantly lower than predicted [13]. These observables are almost free of theoretical uncertain-ties since the hadronic uncertainuncertain-ties cancel out to a large extension in the ratio. While these results still need to be confirmed by an independent experiment (Belle-II), they triggered an intense theoretical activity to understand the source of these discrepancies. Within the 2HDM scenarios discussed above, the violation of LFU is found to be negligibly small, sug-gesting that other bosonic contributions are needed to accommodate these discrepancies. To this purpose, we consider the scenarios containing various LQ states. While these parti-cles are often considered to be exotic in the direct searches at the LHC, they offer one of the prominent candidates to explain the effects of LFU violation. In this thesis, we will scru-tinize the proposed LQ explanations of Rexp

K(∗) < R

SM

K(∗) and discuss the implications for the

current and future experiments. In particular, we will show that a popular scenario in the literature is not viable, and we will propose a new LQ mechanism to explain Rexp

K(∗) < RSM_K(∗)

through loops. Among the predictions of these models, we will emphasize the importance of lepton flavor violating (LFV) decays, since they offer a very clean alternative allowing to test most of the proposed New Physics scenarios. Another intriguing evidence of LFU violation was unveiled in the processes mediated by the charged current [14], where it was found that RD(∗) = B(B → D(∗)τ ν)/B(B → D(∗)lν), with l = e, µ, are larger than predicted

in the SM [15–17]. The possibility that the b → s and b → c anomalies are generated by the same mechanism, possibly related to flavor breaking effects beyond the SM, triggered a lot of interest in the theory community. One should, however, be cautious because the prediction of RD∗ requires the assessment of the B → D∗ form factors which are still not

available from first principle computations. In this thesis, after critically reviewing the status of the SM predictions of RD(∗), we will discuss the models that have been proposed

to simultaneously explain the ensemble of LFU violating observables. In particular, we will discuss a minimal LQ model that we proposed to explain some of these deviations.

The outline of this thesis is as follows. In Chapter1, we briefly introduce the SM to fix our notation and we discuss some of the SM problems which suggest the existence of NP. In Chapter 2, we present the flavor observables that will be discussed in the subsequent chapters. In particular, we will overview the assessment of hadronic uncertainties entering these quantities and give the general expressions for the relevant observables in terms of an effective field theory. In Chapter 3, we present the general features of two-Higgs doublet models and discuss the most relevant probes of the additional scalar particles at low energies. In Chapter4, we introduce the main generalities of leptoquark models and discuss how these particle can arise from grand unification models. In Chapter 5, we discuss the tantalizing hints of NP in lepton flavor universality ratios of B-meson decays. We attempt to overview the viable NP explanations to these puzzles which have been proposed in the literature. In particular, we will focus our discussion on the leptoquark models introduced in Chapter 4. Finally, we summarize our results and conclude.

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The Standard Model

The Standard Model (SM) of particle physics is one of the most successful physical theories ever concieved. Its predictions have been experimentally tested in a wide range of energy scales and they agree very well with the data. Furthemore, the Higgs boson found at the LHC in 2012 is the last missing piece of the SM which corroborated more than five decades of phenomenological success [1,2]. Nonetheless, there are several phenomenological and aesthetic reasons which indicate that the SM cannot be the ultimate fundamental theory.

The purpose of this Section is to fix our notation by introducing the SM Lagrangian and to briefly discuss the limitations of the SM which motivate the quest for physics beyond the SM.

1.1 The Standard Model Lagrangian

The SM is a quantum field theory that describes the fundamental electromagnetic, weak and strong interactions based on the gauge principle. The building blocks of the SM are fermions (leptons and quarks), which appear in three copies of chiral multiplets. The interactions in the SM are introduced by the gauge group

GSM= SU(3)c× SU(2)L× U(1)Y , (1.1)

where SU(3)cis the group associated to the Quantum Chromodynamics (QCD), SU(2)Lto the weak isospin, and U(1)Y the hypercharge. To give masses to the SM particles, a scalar field belonging to a SU(2)L doublet (Φ) is also introduced. This doublet is responsible for the spontaneous symmetry breaking GSM→ SU(3)c× U(1)emvia the so-called Higgs-Brout-Englert mechanism [18–21].

The quantum numbers of the SM fields are summarized in Table 1.1. By imposing the requirements of renormazability and gauge invariance, one can write the most general Lagrangian for the SM in the following form

LSM= Lgauge+ Lmatter+ LHiggs+ LYukawa. (1.2) The expressions for each piece in LSM will be described below.

The gauge sector

The SM gauge sector contains eight gluons Ga

µ (a = 1, . . . , 8), three electroweak gauge bosons Wa

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SU(3) SU(2) U(1)Y Li 1 2 1/2 Qi 3 2 1/6 `R 1 1 −1 uR 3 1 2/3 dR 3 1 −1/3 Φ 1 2 1/2

Table 1.1: Representations under which the SM fermion and scalar fields transform. For the non-Abelian groups, the representation is denoted by its dimension.

fields transform under the adjoint representation of the corresponding gauge groups. The gauge Lagrangian is then described by

Lgauge= −1 4GaµνG µν,a₋ 1 4Wµνa W µν,a₋1 4BµνBµν, (1.3)

where the summation over the SU(N) indices is implicit, a = 1, . . . (N2 ₋1). The field strength tensors are defined by

Ga_µν = ∂µGa_ν − ∂νGµa+ gsfabcGbµGcν, (1.4) W_µνa = ∂µW_νa− ∂νWµa+ g ε abc_Wb µW c ν , (1.5) Bµν = ∂µBν− ∂νBµ, (1.6)

where εabc and fabc are the structure constants of the SU(2)L and SU(3)c groups, respec-tively. Moreover, g and gs are the couplings of SU(3)c and SU(2)L, respectively. For completeness, we also define the coupling g0 _{of the Abelian group U(1)Y}_{. The mass terms} in Lgauge are forbidden by the gauge symmetry. For the electroweak gauge bosons W± and Z, this mass is generated by the Higgs-Brout-Englert mechanism, which will be briefly described below.

The scalar sector

To generate the masses of the gauge bosons, the gauge symmetry must be broken. The main idea of the Brout-Englert-Higgs mechanism is that the dynamics of the theory can be such that a symmetry of the Lagrangian is not necessarily respected by its ground state [18–21]. In this case, the choice among one of the possible degenerate vacuum states spontaneously breaks the symmetry.

In the SM, the most general Lagrangian in the scalar sector reads LHiggs= (DµΦ)

†_{(DµΦ) + m}2_Φ†_{Φ − λ} Φ†_Φ2

, (1.7)

where m2 _{and λ are free parameters. The quartic coupling λ is positive, since the scalar} potential must be bounded from below. The covariant derivative of the Higgs doublet is

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defined by DµΦ = " ∂µ− ig σa 2 Wµa− i g0 2Bµ # Φ , (1.8)

where σa _{(a = 1, 2, 3) are the Pauli matrices. In the case m}2 _<_{0, the minimization of the} scalar potential leads to the ground state hΦ†_{Φi = 0, which is a singlet of the SM gauge} symmetry. To implement the Brout-Englert-Higgs mechanism, one must assume m2 _> 0, in which case the minimization of the Higgs potential leads to a non-trivial vacuum,

hΦ†Φi = v 2

2 , (1.9)

where v = m/√λ is the vacuum expectation value (vev), which induces the spontaneous

breaking of the SM symmetry SU(3)c× SU(2) × U(1)Y into SU(3)c× U(1)em. Among the possible vacuum states, we can choose

hΦi = √1 2   0 v   . (1.10)

The scalar doublet Φ had four degrees of freedom, three of which are the Goldstone bosons eaten up to give mass to the electroweak bosons. The remaining scalar particle is the so-called Higgs boson h, which was discovered in 2012 at the LHC [1,2]. In the unitary gauge, the scalar doublet Φ can be parameterized as

Φ = √1 2   0 v+ h   . (1.11)

By inserting Eq. (1.11) in Eq. (1.7), one can then recognize three massive gauge bosons W± and Z0_{, defined by} W_µ± = W 1 µ_√∓ iWµ2 2 , (1.12) and    Zµ= Wµ3cos θW + Bµsin θW , Aµ= −Wµ3sin θW + Bµcos θW, (1.13) with masses mW = gv 2 , and mZ = gv 2 cos θW . (1.14)

In addition to the W± _{and Z}0, there is one massless vector field Aµ which is related to the photon. The fact that the photon is massless is a manifestation of the residual symmetry

U(1)em of the theory. The Weinberg angle θW is defined by

tan θW = g0

g , (1.15)

which is related to the electric charge via the relation

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On the phenomenological level, the electroweak vev can be determined from the muon lifetime, which is related to the Fermi constant GF = 1.1663787(6) × 10−5 _GeV−2 _[₂₂_{] via} the relation

v2 = √1

2GF . (1.17)

This fixes the vev to be v ≈ 246 GeV. The other parameters in the scalar sector are fixed by the Higgs boson mass mh = √2m2 = 125.09(24) GeV [22]. This value for mh is in good agreement with the indirect bounds1 from the LEP data [24] and implies that λ = m2_h/(2v2) ≈ 0.13 for the quartic coupling, which is therefore perturbative. Furthermore,

the relations between the different couplings and masses of the electroweak bosons are one of the main predictions of the SM, which have been thoroughly verified at different experiments, including LEP, Tevatron and the LHC. The current experimental averages for the W± _{and Z}0 _{masses are given by [}₂₂_]

mW = 80.385(15) GeV ,

mZ = 91.1876(21) GeV .

(1.18) These values are found to be consistent with the global fit to the electroweak observables, to which the radiative corrections must be included. The assessment of the validity of the SM at electroweak scale via electroweak precision tests, including the parameters S, T and

U which permit to test it at the loop-level, is one of the most important achievements of

the SM [23].

Fermionic sector

The kinetic and gauge interactions of fermions are given by Lmatter = X i=e,µ,τ iLiDL/ i+ X i=e,µ,τ i`RiD`/ Ri+ X quarks iQ_iDQ/ i+ X quarks iq_RiDq/ Ri, (1.19) where the summations extend over the different fermions appearing in Table 1.1 and over family indices. Fermion masses are forbidden at this level, arising only in the Yukawa sector via the Higgs mechanism, as it will be discussed in the following. The covariant derivatives for the weak doublets are defined by

DµψL= " ∂µ− ig √ 2 τ+W_µ++ τ−W_µ−+ ieQAµ− ig 2 cos θW τ3−2Q sin2θW Zµ # ψL, (1.20) where we have used Eqs. (1.12) and (1.13) to make explicit the physical gauge bosons. In this expression, Y is the hypercharge of the fermion multiplet and the charge operator is defined as Q = Y + T3. Moreover, we defined τ± _{= (τ1}_{± iτ}_{2)/2, as usual. For RH fermions,} which are SU(2)L singlets, the covariant derivative takes a simpler form, namely,

DµψR= " ∂µ+ ieQAµ+ ig cos θWQsin2θWZµ # ψR. (1.21)

From the expressions given above, one can derive the form of charged and neutral current interaction in the flavor basis.

1_{The SM fit to electroweak data predicts m}

h= 94+25−22 GeV [23], which agrees within 1.3σ with the value measured at the LHC.

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Yukawa sector

In the SM, the only source of flavor violation are the Yukawa couplings between fermions and the Higgs field, defined by

LYukawa = −(Y`)ijLiΦ `Rj−(Yd)ijQiΦ dRj−(Yu)ijQiΦ uRje + h.c. , (1.22) where Φ = iσ2Φe ∗ is the conjugate SU(2)L doublet and Yu,d,` are 3 × 3 complex matrices in flavor space. After spontaneous symmetry breaking, the Yukawa Lagrangian in the unitary gauge becomes

LYukawa = −h√+ v 2

"

(Y`)ij`Li`Rj+ (Yd)ijdRidRj+ (Yu)ijuLiuRj #

+ h.c. , (1.23) from which one can read the interactions of the Higgs boson to fermions and the charged fermion masses, namely,

Mf =

Yfv √

2 , (1.24)

where f = u, d, `. These fermion mass matrices are non-diagonal and can then be diago-nalized by biunitary transformations,

Mf = VLf †M diag

f V

f

R. (1.25)

The unitary matrices Vf

L and V f

R can be absorbed by a redefinition of the LH and RH fermion fields,

Ψf,L → V_Lf †Ψf,L, Ψf,R→ V_Rf †Ψf,R. (1.26)

The neutral currents remain flavor diagonal under this transformation, while the charged ones become flavor violating,

Lcc= − g √ 2 X i,j (VCKM)ijuiγµPLdjWµ++ h.c. , (1.27) where VCKM = Vu L V d †

L is the Cabibbo–Kobayashi–Maskawa (CKM) matrix. The CKM matrix is unitary and it can be fully parameterized by three angles and one complex phase, which must be extracted from experiment, similarly to the fermion masses. To see that, note that any n × n unitary matrix has n2 _{parameters, from which n(n − 1)/2 are real and}

n(n + 1)/2 are complex. Some of the complex parameters can be reabsorbed by rephasing

the quark fields. More specifically, by redefining the n down-type quarks and the n up-type quarks, while imposing baryon number conservation, one can eliminate 2n − 1 phases (the relative phases of the quark fields). Therefore, by taking n = 3 we obtain that the CKM matrix has n(n − 1)/2 = 3 angles, and (n − 1)(n − 2)/2 = 1 phase. The observation that a third generation is needed in order to have CP violation in the SM is known as the Kobayashi–Maskawa mechanism [25].

One of the main peculiarities of the SM is that flavor changing neutral currents (FCNC) are absent at tree-level. These processes are only generated at loop-level, as illustrated in Fig. 1.1. Since these phenomena are loop-induced, they are rare in the SM, which is why they provide very useful tests of the validity of the SM and a probe of physics beyond

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the SM, as we will discuss in Chapter 2. Furthermore, it should be stressed that FCNC processes are directly related to fermion masses and mixing because the Yukawa couplings are the only sources of flavor violation in the SM.

q

_i

W

¯

q

_j

q

¯

k

q

1.2 Why do we need to go beyond?

Despite its remarkable phenomenological success, the SM cannot be a final theory of fun-damental interactions. Firstly, it does not incorporate gravitational interactions, which should become important at energy scales near the Planck scale. The solution to this long standing problem is one of the great open questions in theoretical physics and the lack of experimental signals to guide us makes this task particularly difficult. In addition to the lack of a quantum theory of gravity, the SM cannot accommodate some experimental observations, such as neutrino masses and the observation of dark matter. While there is no doubt that neutrinos are massive particles, the conclusion that dark matter must man-ifest itself as new particles interacting weakly with the SM depends on the validity of the Standard Model of cosmology. This inference must be corroborated by direct and indirect searches for dark matter particles, which have been inconclusive until the present moment. Moreover, in additional to neutrino masses, the SM also leaves many aesthetic questions unanswered, such as the hierarchy and flavor problems.

In the following, we will briefly overview some of the above mentioned problems which motivate physics beyond the SM.

1.2.1 Neutrino masses

One of the most compelling evidences of physics beyond the SM comes from the observation that neutrinos are massive particles that oscillate among different flavors. The picture of massive neutrinos has been corroborated by several experiments measuring atmospheric [26– 31] and solar neutrinos [32,33,33–41], as well as neutrinos produced in accelerators [42–44] and nuclear reactors [45–48]. In 2012, the last missing angle in the lepton sector, θ13 ≈8◦, was measured by reactor experiments [49], which helped establishing that neutrinos are indeed massive and oscillate. In addition to the mixing angles, the oscillation experiments

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can also measure the squared mass differences ∆m2

ij = m2νi − m

2

νj (i, j = 1, 2, 3) of the

neutrinos mass eigenstates νi, which satisfy [50] ∆m2 21 ≈7.5 × 10 −5 _eV2 , and ∆m 2 3` ≈2.5 × 10 −3 _eV2 . (1.28)

The ordering of neutrino masses is still unknown, which can be normal mν1 < mν2 < mν3,

or inverted mν3 < mν1 < mν2. For that reason, we define in the above equation ∆m

2 3` = ∆m2

31 >0 for the normal ordering, and ∆m23`= ∆m232<0 for the inverted one.

The absolute mass scale of neutrinos remains unconstrained, since the oscillation proba-bilities are only sensitive to the squared mass differences. Limits on mνi can be obtained by

studying the kinematics of tritium beta decay, which sets limits of the order mνi . 1 eV [51].

Stronger limits on the sum of neutrino masses can be obtained from cosmology, but these require further assumptions about the cosmological model. For this reason, those bounds are less compelling. Based on the discussion above, it becomes clear that neutrinos are not only massive fermions, but also that they are several orders of magnitude lighter than the other SM particles. This mass gap is often considered as an indication that neutrino masses might be generated by a different mechanism than the one inducing the charged fermion masses.

On theory side, the SM is constructed in such a way that neutrinos are massless, as discussed Sec. 1.1. Therefore, a mechanism beyond the SM is needed to generate neutrino masses. It is beyond the scope of this thesis to review the plethora of scenarios proposed to generate neutrino masses and to explain their smallness. Nonetheless, it is fair to say that the minimal scenarios predict the existence of particles well beyond the reach of current and future experiments, see for example the original proposal of the type I seesaw mecha-nism [52–55]. To lower these energy scales, it is usually necessary to introduce new degrees of freedom, at the price of increasing the complexity of the models, as in the so-called Inverse seesaw scenario [56–58]. Other possibility is to fine tune the couplings. Finally, in terms of an effective theory, the lowest-dimension operator capable of generating neutrino masses is the Weinberg operator [59],

O5 = fij 2Λ LC i Φf∗ f Φ†_L j + h.c. , (1.29)

which is the only dimension-5 allowed by the SM gauge symmetry. In this equation, fij are complex numbers satisfying fij = fji and Λ is an energy scale associated to the mechanism behind neutrino masses. Moreover, the fermion conjugation of a generic fermion field Ψ is defined by ΨC _{= γ0}_C_Ψ∗_{, where C = iγ}2_γ0_{. The Weinberg operator violates lepton number}

L by two units and gives the following Majorana mass term

L ⊃ fijv 2

4Λ νLiC νLj+ h.c. (1.30)

It should be stressed that any scenario of NP generating Majorana neutrino masses can be matched onto an effective theory containing O5 once the heavy degrees of freedom have been integrated out.

The mechanism just described can generate neutrino masses if the lepton number sym-metry L is violated, i.e. if neutrinos are Majorana particles. Until now, the violation of L has still not been observed in experiments, which have been mostly focused on the search of the neutrinoless double beta decay nn → ppe−_e− _[₆₀_–₆₂_{]. Therefore, despite the strong} theoretical preference for Majorana neutrinos, the hypothesis of Dirac neutrinos remains equally plausible. If neutrinos are Dirac particles then the generation of their masses would

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proceed through the Yukawa interactions with the introduction of right-handed neutrinos, in a similar to way to the mechanism for charged fermions.

1.2.2 The SM flavor problem

The origin of the flavor of quarks and leptons remains one of great open questions in theoretical physics. The only source of flavor violation in the SM comes from the Yukawa interactions, which break the global flavor symmetry of the SM

GG = U(3)L× U(3)`R × U(3)Q× U(3)uR × U(3)dR, (1.31)

where each U(3) transformation acts in flavor space for a fermion field in Table 1.1. The description of flavor in terms of the Higgs mechanism is highly unsatisfactory since the Yukawa interactions are not controlled by any symmetry principle, contrary to the SM gauge sector, resulting in a large number of parameters that must be extracted from experiment. In other words, the SM Higgs sector can only accommodate fermion masses and mixing, but it does not constraint the size or pattern of Yukawa couplings.

The flavor problem arises when one inspects the very hierarchical pattern taken by the flavor violating parameters, namely fermion masses and CKM entries. The observed pattern does not look accidental and it is expected to be explained by a symmetry argument or a flavor theory based on that symmetry. If for the moment we put aside θQCD, in the SM there are 13 free parameters in the SM flavor sector, which comprise masses, mixing angles and one complex phase. Charged fermion masses span six orders of magnitude, going from the very light electron with mass me = 0.5110 MeV to the top quark mass

mpolet = 173.3 ± 0.8 GeV [63], which is the heaviest particle of the SM. This hierarchy becomes even more pronounced when one considers neutrino masses, which are known to be mνi . O(1 eV), as depicted in Fig. 1.2. Up to now, there is no convincing explanation

of that hierarchy.

Figure 1.2: Fermion masses in the SM ammended with neutrino masses. We consider the MS quark masses from Ref. [22] at the scale µ = 2 GeV for light quarks, and at µ = mq for q = c, b, t. Another facet of the flavor problem comes from the mixing parameters. CKM and PMNS matrices have very different structures: while CKM is hierarchical, PMNS is not.

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The CKM matrix can be generically parameterized by three angles θ12, θ13 and θ23, and one phase δ13 [22], VCKM=    c12c13 s12c13 s13e−iδ13 −s12c23− c12s23s13eiδ13 c12c23− s12s23s13eiδ13 s23c13 s12s23− c12c23s13eiδ13 −c12s23− s12c23s13eiδ13 c23c13    , (1.32)

where sij ≡ sin θij and cij ≡ cos θij (i, j = 1, 2, 3). The off-diagonal elements of the CKM matrix are found to satisfy a strong hierarchical pattern: |Vus|and |Vcd|are of order ≈ 0.22, |Vcb| and |Vts| are of order ≈ 4 × 10−2, whereas the elements |Vub| and |Vtd| are of order ≈ 5 × 10−3 _[₆₄_,₆₅_{]. This hierarchical pattern becomes even more explicit in the so-called} Wolfenstein parameterization, which makes manifest the fact that s12  s23  s13 [66]. This parameterization is obtained by expanding the CKM matrix is expanded in powers of the small parameter λ ≡ |Vus|≈0.22 1,

VCKM=      1 −λ2 2 λ A λ 3_{(ρ − iη)} −λ 1 −λ2 2 A λ 2 A λ3(1 − ρ − iη) −A λ2 1      + O(λ4_{) ,} _(1.33) where A, ρ and η are O(1) real numbers.2 The present fits to the ensemble of flavor changing

processes result in the precise values ρ = 0.157(14), η = 0.352(11), A = 0.833(12) and

λ = 0.22497(69) [64]. On the other hand, the leptonic mixing matrix, also called

Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, shows a completely different pattern with very large mixing angles. The most recent fits to neutrino data give, to 1σ accuracy [50],

θ₂₃PMNS= (41.6+1.5−1.2)◦, (1.34)

θ₁₂PMNS= (33.56+0.77−0.75)◦, (1.35)

θ₁₃PMNS= (8.46 ± 0.15)◦, (1.36)

where we have used the same parameterization of Eq. (1.32) and taken the results for the normal ordering for illustration. For comparison, the largest angle in the CKM matrix is given by the Cabibbo angle, θCKM

12 = (13.00 ± 0.04)◦, which is of the same order as the smallest mixing angle in the lepton sector, θPMNS

13 = (8.46 ± 0.15)

◦_{. Why are the CKM} parameters so hierarchical? Why are the mixing parameters so large in the leptonic sector? These questions call for physics beyond the SM.

The ensemble of striking observations described above is called the flavor problem, which requires physics beyond the SM. The fact that there is no theoretical hint for the scale of NP responsible for the flavor breaking mechanism beyond the SM makes the flavor problem particularly difficult to solve. Therefore, experimental hints are more than needed to guide the theoretical efforts towards a flavor theory that could give an explanation for the observed patterns of flavor parameters by using symmetry principles.

2_{The inclusion of terms O(λ}4_{) and O(λ}5_{) is mandatory for phenomenological applications.} _The expansion at higher orders can be obtained by adopting the convention λ ≡ s12, Aλ2 ≡ s23 and

Aλ3_{(ρ − iη) ≡ s} 13e−iδ.

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1.2.3 The hierarchy problem

The Higgs sector is the least constrained piece of the SM Lagrangian by symmetries, being for that reason the source of many of its puzzles. We have already discussed the SM flavor problem, which is related to the lack of a flavor symmetry in the Yukawa sector (Higgs couplings to fermions). The hierarchy problem is related to the fact that the SM contains a fundamental scalar with a mass term unconstrained by symmetries, which receives then large radiative corrections from the UV completion of the theory.

In the SM, the masses of fermions and vector bosons are forbidden by the SU(2)L×U(1)Y gauge symmetry. In this case, it can be shown that any loop correction to these parameters will be proportional to the tree-level masses, which means that they are multiplicatively renormalized. This property holds for fermions even in the absence of the gauge symme-try, because the chiral symmetry also protects fermion masses. For the Higgs boson, the situation is different since the limit mh → 0 does not enhance the SM symmetries. In this case, we say that the parameters m2

h is additively renormalized since it receives corrections proportional to the masses of the particles running in the loops. As a consequence of that, the SM Higgs sector becomes highly sensitive to quantum corrections and to the cutoff of the theory.

To illustrate the above-mentioned issue, one can compute the one loop-corrections to the Higgs mass in the SM, which are illustrated in Fig. 1.3. By regularizing the integral with a hard cutoff Λ, one obtains

δm2_h = 3Λ 2 8π2_v2 " 4m2 t −2m 2 W − m 2 Z − m 2 h+ O log Λ µ ! # , (1.37)

where we recognize contribution from top quark which is proportional to m2

t/v2, and that gives the most significant radiative correction to the Higgs mass in the SM. From this expression we see that the Higgs mass is quadratically sensitive to the cutoff of the theory. Since the only available cutoff is the Planck scale, MPl ≈ 1.22 × 1019 _{GeV, which is much} larger than the Higgs mass mh = 125.09(24) GeV [22], we observe that the bare Higgs mass and its counter term must be fine tuned to an enormous accuracy to reproduce a light Higgs mass. This statement is precisely the hierarchy problem, which arises in theories with fundamental scalars where a hierarchy of scales mh Λ is present. In particular, the hierarchy problem is an indication that the NP scale cannot be too large. More precisely, for the Higgs mass observed at the LHC, the requirement of fine tuning in mh of less than 1 part in 10 implies that the scale of NP should be below a few TeV [67].

t t W, Z W, Z h h W, Z h

Figure 1.3: One loop corrections to the Higgs self-energy in the SM.

We have not yet enough information to unambiguously determine how the SM should be extended to address the hierarchy problem. Most concrete proposals point at the existence of new degrees of freedom in the TeV range, possibly accessible at the LHC. This observation is in apparent contradiction with the information extracted from the flavor experiments, which set stringent bounds on the NP scale ΛNP. More specifically, FCNC processes in

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the SM are suppressed by (i) the loop factors, (ii) by the GIM mechanism [68] and (iii) by the hierarchy of the CKM matrix, |Vtd| |Vts| |Vtb|≈ 1. Hence, the precise study of these processes and the comparison with the experimental results allows us to impose severe constraints on ΛNP. For instance, the K0_{− K}0 mixing parameter K sets an lower bound of about ΛNP & 108 GeV if O(1) NP couplings are assumed [6]. This apparent contradiction between bounds on NP stemming from flavor experiments and the prejudice that NP should emerge at the TeV scale is often called the NP flavor puzzle, and it implies that NP should have a non-trivial flavor structure. The most popular solution to this problem goes under the name of Minimal Flavor Violation (MFV) [69,70]. In this approach, the Yukawa interactions are identified as the only source of flavor violation, reproducing the SM flavor structure also beyond the SM. In this case, the bounds from flavor changing observables are lowered to the TeV range at which we expect that the solution to the hierarchy will emerge. Nonetheless, it is worth stressing that the MFV hypothesis is not a theory of flavor, since it does allow us to determine the pattern of Yukawa couplings in terms of a small number of parameters of a more fundamental theory. Therefore, the ultimate solution to the hierarchy and flavor problems remains an open question. Experimental hints from indirect and/or direct searches are in that respect most welcome guides of the theoretical effort to the solution of these problems.

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Flavor observables as a probe of New

Physics

The current situation of direct searches of New Physics is that no particle beyond the SM has been found at the LHC at energies . 1 TeV, and that most theoretical speculations about the possible scale/nature of New Physics seem to be unsound. In such a situation, flavor physics can be very useful, since it can probe scales well above the ones attained by the direct experimental searches. Furthermore, the constraints coming from indirect searches can be complementary in guiding the direct searches, pointing to the observables where New Physics could be effectively seen.

The main difficulty in the comparison between the SM predictions and the experimental results lies in the fact that non-perturbative QCD remains not under full theoretical control. While an analytic solution to non-perturbative QCD is still lacking, in some situations the hadronic uncertainties can be controlled by means of numerical simulations of QCD on the Lattice (LQCD). Over the past decades we witnessed a huge progress of LQCD simulations which nowadays allows us to attain a precision at the percent level for certain hadronic quantities. At the same time, the experimental precision for many observables will be substantially improved by the new generations of flavor experiments. For these reasons, indirect searches and in particular those involving flavor physics are a very promising route to seek the effects of physics beyond the SM.

The goal of this chapter is to review the flavor physics results after the first run of LHC, and the results from meson factories (CLEO-c, BES, BaBar and Belle, among others), which experimentally established the unitarity of the CKM matrix with great precision. In this Chapter, our approach will rely on Effective Field Theories (EFT), since they provide the most general description of low-energy physics in the absence of new light degrees of freedom. Particular emphasis of this Chapter will be given to the assessment of theoretical uncertainties (mainly those of non-perturbative QCD) entering the SM predictions for the flavor physics observables.

The chapter is organized as follows: In Sec.2.1, we briefly introduce the concept of EFT. In Sec. 2.2, we discuss tree-level electroweak decays of mesons and how those can be used to search the search the effects of New Physics. In Sec. 2.3, we extend our discussion to the FCNC processes, and most particularly to the transition b → s`+_`−_{. Lastly, we conclude} the Chapter in Sec.2.4 with a discussion of lepton flavor violation in decays of both leptons and hadrons.

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2.1 Effective field theories

Effective field theories (EFT) are nowadays widely used in all branches of theoretical physics. The basic principle which warranties the validity of EFT method is that one can isolate sets of phenomena with different energy/length scales and then find a suitable description for each class of physical problems. This property is greatly responsible for the gradual progress in physics, where one could for instance formulate the Newtonian mechanics well before the discovery of special relativity, or understand quantum mechanics without having to deal with the uneasy subtleties of QFT.

In the context of QFT, the relevant parameter is the distance scale. Short distance contributions are replaced by local operators, which are obtained by integrating out the irrelevant degrees of freedom, and by expanding in a suitable chosen expansion parameter. The small finite size effects are then treated as perturbations. In particular, this approach is useful for studying the effects of physics beyond the SM in a model independent manner. More precisely, in the so-called bottom up approach, one can write the most general effective Lagrangian consistent with the symmetries of the low energy theory which contains only the light degrees of freedom, i.e.

Leff = Ld=4+ X

i

Ci(µ)

Λdi−4Oi(µ) , (2.1)

where Oi are operators of dimension di > 4, µ is the renormalization scale, and Ci(µ)

are the so-called Wilson coefficients. The Lagrangian Ld=4 describes the (renormalizable) low energy theory, which is typically considered to be the SM. Even though Leff is non-renormalizable, the predictiviy of the effective theory is guaranteed by the fact that the contributions of higher dimensional operators to the amplitudes are suppressed by powers of (p/Λ)di−4, where p is the typical momentum of the low energy process. Henceforth, by

working in the regime p Λ and to a given experimental precision, one can renormalize the effective theory with a finite number of counter-terms. The effects of the (unknown) short distance physics are then encoded in the Wilson coefficients Ci(µ), which can be matched onto a full theory by requiring that the amplitudes computed in the full and effective theories coincide at large distances. The main advantage of this approach is that one can generically describe the low-energy physics without having to postulate what happens at arbitrarily high energy scales.

2.2 Tree-level electroweak decays of mesons

Tree-level electroweak decays of kaons, and D(s) and B mesons are induced by a tree-level exchange of the W boson, as illustrated in the left panel of Fig. 2.1. These processes provide a straightforward way of extracting the moduli of several CKM matrix elements, such as |Vud|, |Vus|, |Vub|, |Vcd|, |Vcs| and |Vcb|, which can then be confronted with the CKM matrix V ≡ VCKM unitarity, V†_{= V}−1_{. To do so, the determination of the hadronic matrix} elements plays a crucial role. Nowadays, the level of maturity of Lattice QCD (LQCD) simulations and the developments on the experimental side allow us to go beyond the simple determination of CKM entries, and to use leptonic and semileptonic decays to probe scenarios of physics beyond the SM. For instance, in scenarios with two Higgs doublets, a charged Higgs H± _{can generate a tree-level contribution to these decays, as depicted} in Fig. 2.1. Therefore, the study of these decays offer a low-energy window to probe the bosonic sector beyond the SM.

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Another motivation to search for NP effects in tree-level semi-leptonic decays comes from the discrepancies observed by the B-physics experiments in the lepton flavor universality ratios, namely

R_D(∗) =

B(B → D(∗)_{τ ν})

B(B → D(∗)_lν) , (2.2)

where l = e, µ are averaged in the denominator. The experimental value of Rexp

D determined by BaBar [71,72] and Belle [73] was found to be about 2σ larger than the SM prediction [14]. This observation was further corroborated by the measurement of Rexp

D∗ [71–76] which

in-dicates a ≈ 3.3σ excess with respect to the quoted SM prediction [17]. If these deviations are indeed generated by NP, it is likely that similar effects are present in other leptonic and semileptonic decays. The experimental and theoretical status of RD(∗) will be discussed in

Chapter 5, where we will also review some of the proposed NP explanations.

B

0

V

cb

D

+

¯

d

b

c

ℓ

−

¯

ν

W

−

B

0

D

+

¯

d

b

c

ℓ

−

¯

ν

H

−

Figure 2.1: Contributions to the semi-leptonic decay B0 → D+_{`ν in the SM (left panel) and in} 2HDM (right panel).

The dimension-6 effective Hamiltonian describing the transition u → d`ν can be gener-ically parameterized as 1 Heff= √ 2GFVud h (1 + gV)(uγµd)(`LγµνL) − (1 + gA)(uγµγ5d)(`LγµνL)

+ gS(µ)(ud)(`RνL) + gP(µ)(uγ5d)(`RνL) + gT(µ)(uσµν(1 − γ5)d)(`RσµννL)i+ h.c.,

(2.3) where and u and d stand for generic up-type and down-type quarks, and ` = e, µ, τ. The NP couplings gV,A,S,P,T are defined relatively to the Fermi constant, namely,

GF √ 2 = g2 8m2 W . (2.4)

Note that the NP couplings can depend on the renormalization scale µ, and that the SM corresponds to gV,A,S,P,T ≡0.

The relevant decay channels can be separated into two classes of processes with different QCD content:

• Leptonic decays: K− _{→ `ν}_{, D}−

(s) → `ν and B → `ν;

• Semileptonic decays: KL→ π±_`ν_{, D → K`ν and B → D`ν, among others,} 1_{We neglect the possibility of RH neutrinos in Eq. (}_2.3_).

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where ` stands for a generic charged lepton. For leptonic decays, the non-perturbative content needed to compute the branching ratios amounts to a single quantity, which is the meson decay constant, computed in numerical simulations of QCD on the Lattice (LQCD). The situation of semileptonic modes is more elaborated, as it involves several q2 dependent form-factors where q2 _{stands for the dilepton mass squared.}

In the following, we will discuss the peculiarities of each of these process, including the non-perturbative inputs needed to do phenomenology, and we will give the general expressions for the experimentally accessible observables in terms of the effective couplings defined in Eq. (2.3).

2.2.1 Leptonic decays of mesons

In this Section, we discuss the decay modes of the type P− _{→ `}_{¯ν, where P = K, D(s)}_{, B} (c) is a pseudoscalar meson (JP _{= 0}−_{). For simplicity, we will write the expressions for the} decays of B mesons, but all the other modes can be obtained mutatis mutandis.

From Lorentz and parity symmetries, we know that only axial and pseudo-scalar hadron currents can contribute to a transition of the type P− _{→ `ν}_{. The most general} parameter-ization of the axial hadronic matrix element reads

h0|¯bγµγ5u|B(p)i = ipµfB, (2.5)

where pµ _{is the 4-momentum of the B meson, and fB} _{is the so-called decay constant. By} virtue of the axial Ward identity, the matrix element of the pseudoscalar density is given by

h0|¯bγ5u|B(p)i = −i m

2 BfB

mu+ mb

. (2.6)

The decay constant fB encapsulates the non-perturbative content of the transition and it has to be computed by numerical simulations of QCD on the lattice. The current values of the different decay constants obtained by LQCD simulations are summarized in Tab. 2.1, which show a degree of precision below the percent level [77].

Quantity Value [MeV]

fK 155.6(4) fD 212.2(15) fDs 249.8(13) fB 186(4) fBs 224(5) fBc 434(15)

Table 2.1: Decay constants of pseudoscalar mesons computed by numerical simulations of QCD on the lattice. These values were extracted from Ref. [77] with the exception of fBc, which was

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By using the Hamiltonian Eq. (2.3) and the hadronic matrix elements defined Eqs. (2.5) and (2.6), one can compute the amplitude for B− _{→ `}_¯ν:

M= −i√2Vub(ifB) ¯u(k1) " (1 + gA)γµ PLpµ+ gP(µ)PL m2 B mu+ mb # v(k2) , (2.7)

where the convention for the momenta of the particles are B−_{(p) → `(k1)¯ν(k2) and the} couplings gAand gP parameterize the most general contributions from NP. Multiplying this amplitude by its conjugate and after summing over the spins we get

X spins |A|2_{= 2G}2 F|Vub|2fB2m2`(m2B− m2`) 1 + gA− gP(µ) m2_B m`(mu+ mb) 2 , (2.8)

so that the final expression for the branching ratio reads

B(B−→ `ν) = G 2 FmBm2` 8π 1 − m2 ` m2 B !2 f_B2|Vub|2τB+ 1 + gA− gP(µ) m2 B m`(mu+ mb) 2 . (2.9)

We recall that the couplings gV, gS and gT do not contribute to this decay, because the corresponding hadronic matrix elements vanish due to parity and Lorentz invariance of QCD. We reiterate that leptonic decays are considered to be extremely clean, since the whole non-perturbative content of these processes are encoded in a single parameter, the decay constant.

2.2.2 Semileptonic P → P

0

decays

In this Section, we discuss the decays of the type P → P0_`ν_{, where P, P}0 _{= K, D(s)}_{, B}

(s), Bc, ηc are pseudoscalar mesons. We focus on the processes involving pseudoscalar mesons since they require a minimal input from non-perturbative QCD. Electroweak processes with vec-tor mesons or baryons depend on many more hadronic form facvec-tors which are more difficult to study in LQCD. For that reason, we will disregard the semileptonic decays to vector mesons and baryons.

In a similar way to the leptonic case, Lorentz invariance and parity guarantee that the transition P → P0 _{only depends on the coefficients gS,V,T}_{. The hadronic matrix element} which appears in the SM amplitude is defined as

hD+_(k)|¯cγµ_{b| ¯}_B_{(p)i =} " (p + k)µ− m 2 B− m2D q2 qµ # f+(q2) + m 2 B− m2D q2 qµf0(q 2_{) ,} _(2.10) where q = p − k is the dilepton 4-momentum, and f+(q2_{) and f0(q}2_{) are the so-called} vector and scalar form factors. Here we focus on the transition ¯B → D+_`_{¯ν, but our results} can be easily translated to the other semileptonic modes. These form factors satisfy the constraint f+(0) = f0(0), a condition which considerably simplifies the determination of these form-factors from LQCD simulations, as well as in the phenomenological analyses of these decays. To demonstrate this property, it is sufficient to parameterize the matrix element in a simplest different way,